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Factoring Special Products in
Difference of Squares
By L.D.
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Problem 1
9x2-4
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Problem 1
9x2 - 4
3x(3x) 2(2)
Both of these can be squared so I will show you what they are squared by under them. To make our problem we will try to fit the formula a2 + b2 = (a + b)(a-b). Since the first one is a2 (9x2), a is 3x, using this way of thinking, I would say that b is 2. Our answer is on the next page.
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Problem 1
9x2 – 4 = (3x + 2)(3x – 2)
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Example Problems
a2 – 81
36m2 – 25
4x2 – y2
a2 + 64
Remember that both must be PERFECT squares.
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Example Problems
a2 – 81 (a + 9)(a – 9)
36m2 – 25 (6m + 5)(6m – 9)
4x2 – y2 (2x + y)(2x – y)
a2 + 64
This cannot be factored since this method doesn’t work with addition problems, only subtraction.
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Mini Lesson
If you feel you are just doing the problems blindly, check them with F.O.I.L. and you will find that two of the numbers cancel out together.
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Problem 2
2a2 – 200
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Problem 2
2a2 – 200
To make this problem work so that we have squares we will have to divide it by 2.
a2 – 100
Now we can solve that to get (a + 10)(a – 10). We add the 2 back by placing it next to the problem for multiplication, making our final answer look like the slide on the next page.
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Problem 2
2a2 – 200 = 2(a + 10)(a – 10)
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Problem 3
-4c2 + 36
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Problem 3
-4c2 + 36
To make this work we will remember what we did in the last problem and divide the problem by -4, making it c2 - 9. Solving this the normal way we will get
(c + 3)(c-3) which will change to be -4(c + 3)(c-3).
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Problem 3
-4c2 + 36 = -4(c + 3)(c-3)
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Formula
a2 + 2ab + b2 = (a + b)(a + b) or (a + b)2
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Problem 4
25x2+ 10x + 1
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Problem 4
25x2+ 10x + 1
We will answer this using the formula on slide 13.
a2 + 2ab + b2 = (a + b)(a + b) or (a + b)2
25x2+ 10x + 1
We will first get the square root the 25x2 (5x) and place it in the ‘a’ place of the formula. Then we will get the square root of 1 (1) and place it in the ‘b’ place of the formula. The answer will be on the next slide.
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Problem 4
25x2+ 10x + 1 = (5x + 1)2
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Problem 5
144y2 - 120y + 25
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Problem 5
144y2 - 120y + 25
We need to find a way to accommodate the negative sign in the middle so just blindly using our formula to get (12y + 5)(12y + 5) won’t work. We can however, make it (12y - 5)(12y – 5), which will achieve our goals perfectly.
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